The classification of measures using the classical power-law Hausdorff measures and dimensions has found many applications within spectral theory ([1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]} and others). While this classification theory has been very useful in many situations, notably when the Hausdorff dimension is positive, it has not been general enough to understand the differences between zero-dimensional spectra. This has been explored in recent papers by Mavi [11]}, who studied logarithmic dimension bounds for the disordered Holstein model, and Avila, Last, Shamis, and Zhou [12]}, who studied the modulus of continuity of the integrated density of states for the almost Mathieu operator using a logarithmic dimension.
